A151348 Number of walks within N^2 (the first quadrant of Z^2) starting and ending at (0,0) and consisting of n steps taken from {(-1, -1), (-1, 0), (-1, 1), (0, -1), (1, 0)}.

1, 0, 1, 1, 4, 7, 25, 64, 201, 612, 1961, 6355, 21026, 70968, 241810, 837191, 2925393, 10334302, 36813216, 132242756, 478470272, 1742816732, 6387201912, 23539830561, 87207544029, 324627673245, 1213820275167, 4557447698656, 17177881979810, 64981216839823, 246648317043660, 939184339480746, 3586940782960596

Offset

0,5

Links

Robert Israel, Table of n, a(n) for n = 0..1000

A. Bostan, K. Raschel, and B. Salvy, Non-D-finite excursions in the quarter plane, J. Comb. Theory A 121 (2014) 45-63, Table 1 Tag 21, Tag 33.

M. Bousquet-Mélou and M. Mishna, 2008. Walks with small steps in the quarter plane, ArXiv 0810.4387.

Maple

Steps:= [[-1, -1], [-1, 0], [-1, 1], [0, -1], [1, 0]]:
f:= proc(n, p) option remember; local t, s;
      if max(p) > n then return 0 fi;
      add(procname(n-1, s), s = select(t -> min(t)>=0, map(`+`, Steps, p)))
end proc:
f(0, [0, 0]):= 1:
map(f, [$0..40], [0, 0]); # Robert Israel, Nov 20 2023

Mathematica

aux[i_Integer, j_Integer, n_Integer] := Which[Min[i, j, n] < 0 || Max[i, j] > n, 0, n == 0, KroneckerDelta[i, j, n], True, aux[i, j, n] = aux[-1 + i, j, -1 + n] + aux[i, 1 + j, -1 + n] + aux[1 + i, -1 + j, -1 + n] + aux[1 + i, j, -1 + n] + aux[1 + i, 1 + j, -1 + n]]; Table[aux[0, 0, n], {n, 0, 25}]

Crossrefs

Sequence in context: A073218 A276288 A219700 * A211942 A110413 A075686

Adjacent sequences: A151345 A151346 A151347 * A151349 A151350 A151351

Keyword

nonn,walk

Author

Manuel Kauers, Nov 18 2008

Status

approved